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Consider an $N$ by $N$ matrix $X$ of complex entries with iid real andimaginary parts. We show that the local density of eigenvalues of $X^*X$converges to the Marchenko-Pastur law on the optimal scale with probability$1$. We also obtain rigidity of the eigenvalues in the bulk and near both hardand soft edges. Here we avoid logarithmic and polynomial corrections by workingdirectly with high powers of expectation of the Stieltjes transforms. We workunder the assumption that the entries have a finite 4th moment and aretruncated at $N^{1/4}$. In this work we simplify and adapt the methods fromprior papers of G\"otze-Tikhomirov and Cacciapuoti-Maltsev-Schlein tocovariance matrices.

Local Marchenko-Pastur law at the hard edge of the Sample Covariance ensemble